# The Fundamental Theorem of Calculus and Vegetables

When I was an undergraduate, someone presented to me a proof of the Fundamental Theorem of Calculus using entirely vegetables. I found this incredibly fun at the time, but I can't remember who presented it to me and my internet searching has not been successful.

The proof involved pinning various vegetables to a board and using their locations as variable names. I hope to find a reference for this -- obviously it's a bit of a longshot.

I was at the University of Michigan and it probably happened around 2004. But I have no reason to believe it was invented at that location or that time.

• What a title! I thought this was going to be about "eating one's vegetables"…i.e. having to see a proof as a typical Calculus student. Commented Sep 19, 2014 at 11:27
• @BenjaminDickman Ha, no -- I mean literally vegetables, tacked to a board. It may have been original from whoever showed it to us. Commented Sep 22, 2014 at 18:34
• In what way this particular explanation of the fundamental theorem of calculus is important? Does it allow to grasp certain concepts better than other explanations? Are vegetables essential for the explanation, or were used just for fun? How the explanation will change if vegetables are replaced with other objects? I am surprised this question got so many upvotes. Commented Apr 29, 2019 at 17:48
• I assumed at the time of writing that maybe this was a well-known exposition something like Flatland. As it turns out, it wasn't well-known at all and was an idiosyncratic presentation by one specific graduate student. If you don't think it's a good reference-request, feel free to downvote it. Commented Apr 29, 2019 at 20:30
• @FedericoPoloni I found the person I was looking for (5 years later) and hope to have a video to share later this summer. I don't remember the details. Commented May 31, 2019 at 20:40

I am grateful to Mark Conger for finding a video of this presentation and getting the University of Michigan to digitize it. It isn't produced in the style of math youtube content, because that was presumably not the goal of QED TV, which I do not know the history of. But it is found! I finally have a link to the video! Here it is:

• The video spends a lot of time on preliminaries, which you can skip by going to about 13:40.
• Since you have likely proved FTC before, I would recommend you start watching at 17:55, which is when the vegetables appear.

The vegetables are used to represent various things that appear in the proof:

And the final product when h has approached zero (Carrot has approached Asparagus) looks like this:

I remember this anecdotal version, I don't know whether or not it matches yours.

The Fundamental Theorem of Calculus stated using vegetables:

$$\int_{carrot}^{potato}vegetable(turnip)d(turnip)$$ $$=stew(potato)-stew(carrot)$$ where $$stew'=vegetable$$

My tutor said you end up with potato stew with no carrots. I asked what happened to the turnips?

• This does not seem to attempt to answer the question as asked. Please edit your post so that it actually answers the question. (Otherwise this post might be removed.)
– quid
Commented Aug 15, 2015 at 9:18
• @quid; howzat??
– JMP
Commented Aug 15, 2015 at 9:26
• Your edit does help a bit, thanks for that. It still does not seem much like an answer to a reference request for that proof of FTC.
– quid
Commented Aug 15, 2015 at 9:35
• @quid; it might help jog a memory or two...
– JMP
Commented Aug 15, 2015 at 9:36